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Abstract Let S = GR (p e , m) be a Galois ring of characteristic p e and cardinality p e m. An additive code over S of length n is a subgroup of S n under addition. In this paper, we study additive codes over S. We introduce a correspondence between linear codes over Z p e and additive codes over S and we describe additive codes over S by the structure of linear codes over Z p e . In particular, we find the generator matrix and the number of additive codes over S , and we determine some classes of MDR additive codes over S. Among other results, permutation equivalent additive codes and decomposable additive codes are described. Also we prove MacWilliams identity and Delsarte theorem for additive codes over S. [ABSTRACT FROM AUTHOR]

Abstract In this paper some new links between the nonlinearity and differential uniformity of some large classes of functions are established. Differentially two-valued functions and quadratic functions are mainly treated. A lower bound for the nonlinearity of monomial δ -uniform permutations is obtained, for any δ , as well as an upper bound for differentially two-valued functions. Concerning quadratic functions, significant relations between nonlinearity and differential uniformity are exhibited. In particular, we show that the quadratic differentially 4-uniform permutations should be differentially two-valued and possess the best known nonlinearity. [ABSTRACT FROM AUTHOR]

Abstract Let R be a noncommutative ring with unity. The commuting graph of R , denoted by Γ (R) , is a graph whose vertices are noncentral elements of R and two distinct vertices x and y are adjacent if x y = y x. Let F be a finite field and n ≥ 2. It is conjectured by Akbari, Ghandehari, Hadian and Mohammadian in 2004 that if Γ (R) ≅ Γ (M n (F)) , then R ≅ M n (F). In this paper, we prove the conjecture whenever n is of the form 2 k 3 l with k ≠ 0. [ABSTRACT FROM AUTHOR]

Abstract Let q be a power of a prime and F q be a finite field with q elements. In this paper, we propose four families of infinite classes of permutation trinomials having the form c x − x s + x q s , and investigate the relationship between two types of widely discussed permutation polynomials. Based on this relation, many classes of permutation polynomials having the form (x q k − x + δ) s + c x without restriction on δ are derived. [ABSTRACT FROM AUTHOR]

Abstract: The present paper deals with counting the number and orders of regular semisimple elements, together with the number of primary classes of . The approach used in this paper depends essentially on partitions of positive integers ⩽n. We give the numbers of regular semisimple elements and primary classes of for and see that the number of regular semisimple elements is an integral polynomial in q, while the number of primary classes is a rational polynomial in q. [Copyright &y& Elsevier]

Abstract: In this paper, we prove that for any given , there exists a constant such that for any prime power , there exists a primitive normal polynomial of degree n over with the first coefficients prescribed, where the first coefficient is nonzero. This result strengthens the asymptotic result of the existence of primitive polynomials with the first coefficients prescribed [S.Q. Fan, W.B. Han, p-Adic formal series and Cohen''s problem, Glasg. Math. J. 46 (2004) 47–61] in two aspects. One is that we discuss in this paper not only the primitivity but also the normality. Another is that the number of the prescribed coefficients increases from to . The estimates of character sums over Galois rings, the p-adic method introduced by the first two authors, and the computation technique used in [S.Q. Fan, W.B. Han, Primitive polynomial with three coefficients prescribed, Finite Fields Appl. 10 (2004) 506–521; D. Mills, Existence of primitive polynomials with three coefficients prescribed, J. Algebra Number Theory Appl. 4 (2004) 1–22] are the main tools to get the above result. [Copyright &y& Elsevier]

In this paper ideals of certain group algebra are used to prove results on k -normal elements over finite fields, including ways to determine them from normal elements by means of polynomials and circulant matrices. Alternative proofs of results appearing in [4] are also provided. [ABSTRACT FROM AUTHOR]

In this paper we present a family of maximal cliques of size q + 1 2 or q + 3 2 , accordingly as q ≡ 1 ( 4 ) or q ≡ 3 ( 4 ) , in Paley graphs of order q 2 , where q is an odd prime power. After that we use the new cliques to define a family of eigenfunctions corresponding to both non-principal eigenvalues and having support size q + 1 , which is the minimum possible value by the weight-distribution bound. Finally, we prove that the constructed eigenfunction comes from an equitable partition. [ABSTRACT FROM AUTHOR]

In this paper, we characterize all curves over F q arising from a plane section P : X 3 − e 0 X 0 − e 1 X 1 − e 2 X 2 = 0 of the Fermat surface S : X 0 d + X 1 d + X 2 d + X 3 d = 0 , where q = p h = 2 d + 1 is a prime power, p > 3 , and e 0 , e 1 , e 2 ∈ F q . In particular, we prove that any nonlinear component G ⊆ P ∩ S is a smooth classical curve of degree n ⩽ d attaining the Stöhr–Voloch bound # G ( F q ) ⩽ 1 2 n ( n + q − 1 ) − 1 2 i ( n − 2 ) , with i ∈ { 0 , 1 , 2 , 3 , n , 3 n } . [ABSTRACT FROM AUTHOR]

Let p be an odd prime. The main purpose of this paper is to establish and exploit the algebraic structure of some repeated-root two dimensional constacyclic codes of length 2 p s . 2 k over the finite field F p m . Moreover, we study all self-dual repeated-root two-dimensional ( λ , 1 ) -constacyclic codes of length 2 p s . 2 or 2 p s . 2 2 over F p m for λ ∈ { − 1 , 1 } . [ABSTRACT FROM AUTHOR]

Let q be a prime power and F q n be the finite field with q n elements, where n > 1 . We introduce the class of the linearized polynomials L ( X ) over F q n such that L ( t ) ( X ) : = L ∘ L ∘ ⋯ ∘ L ︸ t times ( X ) ≡ 0 ( mod X q n − X ) for some t ≥ 2 , called nilpotent linearized polynomials (NLP's). We discuss the existence and construction of NLP's and, as an application, we show how to obtain permutations of F q n from these polynomials. For some of those permutations, we can explicitly give the compositional inverse map and the cycle decomposition. This paper also contains a method for constructing involutions over binary fields with no fixed points, which are useful in block ciphers. [ABSTRACT FROM AUTHOR]

In this paper we investigate the distribution properties of hybrid sequences which are made by combining Halton sequences in the ring of polynomials and digital Kronecker sequences. We give a full criterion for the uniform distribution and prove results on the discrepancy of such hybrid sequences. [ABSTRACT FROM AUTHOR]

In this paper we present several values for the next-to-minimal weights of projective Reed–Muller codes. We work over F q with q ≥ 3 since in [3] we have determined the complete values for the next-to-minimal weights of binary projective Reed–Muller codes. As in [3] here we also find examples of codewords with next-to-minimal weight whose set of zeros is not in a hyperplane arrangement. [ABSTRACT FROM AUTHOR]

In this paper, by employing some results on Kummer extensions, we give an arithmetic characterization of pure Weierstrass gaps at many totally ramified places on a quotient of the Hermitian curve, including the well-studied Hermitian curve as a special case. The cardinality of these pure gaps is explicitly investigated. In particular, the numbers of gaps and pure gaps at a pair of distinct places are determined precisely, which can be regarded as an extension of the previous work by Matthews (2001) considered Hermitian curves. Additionally, some concrete examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

Relative generalized Hamming weights (RGHWs) of a linear code with respect to a linear subcode determine the security of the linear ramp secret sharing scheme based on the linear codes. They can be used to express the information leakage of the secret when some keepers of shares are corrupted. Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems. In this paper, we investigate the RGHWs of cyclic codes of two nonzeros with respect to its irreducible cyclic subcodes. We give two formulae for RGHWs of the cyclic codes. As applications of the formulae, explicit examples are computed. Moreover, RGHWs of cyclic codes in the examples are very large, comparing with the generalized Plotkin bound of RGHWs. So it guarantees very high security for the secret sharing scheme based on the dual codes. [ABSTRACT FROM AUTHOR]

Let F q denote the finite field of order q . In this paper, some new classes of permutation polynomials of the form ( x p m − x + δ ) s + x over F p 2 m are obtained by determining the number of solutions of certain equations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a class of permutation polynomials over the finite field F 2 2 m for odd m . These permutations are generally quadrinomials, and some permutation trinomials can also be obtained. [ABSTRACT FROM AUTHOR]

In this paper, we present several necessary conditions for the reversed Dickson polynomial E n ( 1 , x ) of the second kind to be a permutation of F q . In particular, we give explicit evaluation of the sum ∑ a ∈ F q E n ( 1 , a ) . [ABSTRACT FROM AUTHOR]

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. Let m = 2 ℓ + 1 for an integer ℓ ≥ 1 and π be a generator of GF ( 3 m ) ⁎ . In this paper, a class of cyclic codes C ( u , v ) over GF ( 3 ) with two nonzeros π u and π v is studied, where u = ( 3 m + 1 ) / 2 , and v = 2 ⋅ 3 ℓ + 1 is the ternary Welch-type exponent. Based on a result on the non-existence of solutions to certain equation over GF ( 3 m ) , the cyclic code C ( u , v ) is shown to have minimal distance four, which is the best minimal distance for any linear code over GF ( 3 ) with length 3 m − 1 and dimension 3 m − 1 − 2 m according to the Sphere Packing bound. The duals of this class of cyclic codes are also studied. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we explore constacyclic self-dual codes over . We first characterize constacyclic self-dual codes over of any length. Then we determine the structure of η-constacyclic self-dual codes over , where or . This structure is used to find some constacyclic self-dual codes over . [Copyright &y& Elsevier]

Abstract: For a plane curve over of degree , it is known by our previous work that the number of its -rational points is at most . In this paper, we determine the curves that attain this maximum, up to projective equivalence. [Copyright &y& Elsevier]

Abstract: In this paper we generalize a recent result of Wittmann on densities of the 4-rank of class groups of imaginary quadratic function fields to the ℓ-rank of class groups of imaginary ℓ-cyclic extensions of where ℓ is any odd prime dividing . [Copyright &y& Elsevier]

Abstract: In this paper, we study rank two semifields of order that are of scattered type. The known examples of such semifields are some Knuth semifields, some Generalized Twisted Fields and the semifields recently constructed in Marino et al. (in press) for . Here, we construct new infinite families of rank two scattered semifields for any q odd prime power, with ; for any , such that and for any with . Both the construction and the proof that these semifields are new, rely on the structure of the linear set and the so-called pseudoregulus associated to these semifields. [Copyright &y& Elsevier]

Abstract: For an odd prime p and two positive integers and k with being odd, the paper determines the weight distribution of a class of p-ary cyclic codes over with nonzeros , and , where α is a primitive element of . [Copyright &y& Elsevier]

Abstract: We investigate the functional code introduced by G. Lachaud (1996) in the special case where X is a non-singular Hermitian variety in and . In , F.A.B. Edoukou (2007) solved the conjecture of Sørensen (1991) on the minimum distance of this code for a Hermitian variety X in . In this paper, we will answer the question about the minimum distance in general dimension N, with . We also prove that the small weight codewords correspond to the intersection of X with the union of 2 hyperplanes. [Copyright &y& Elsevier]

Abstract: In this paper we treat cyclotomic binary duadic codes. The conjecture of Ding and Pless is that there are infinitely many cyclotomic duadic codes of prime lengths that are not quadratic residue codes. We shall prove this conjecture by using the special case of Tschebotareff''s density theorem. [Copyright &y& Elsevier]

Abstract: The semifields of order which are two-dimensional over their left nucleus and six-dimensional over their center have been geometrically partitioned into six classes by using the associated linear sets in . One of these classes has been partitioned further into three subclasses. In this paper the generic multiplication is determined for each of these three subclasses, and several examples of new semifields are constructed that belong to these subclasses. For two of the subclasses, no examples were previously known. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is a description of a model of Kummer surfaces in characteristic 2, together with the associated formulas for the pseudo-group law. Since the classical model has bad reduction, a renormalization of the parameters is required, that can be justified using the theory of algebraic theta functions. The formulas that are obtained are very efficient and may be useful in cryptographic applications. We also show that applying the same strategy to elliptic curves gives Montgomery-like formulas in odd characteristic that are faster than the classical ones, and we recover already known formulas by Stam in characteristic 2. [Copyright &y& Elsevier]

Abstract: Given a set , , is defined as the smallest positive integer k for which . It can be shown that always and if the characteristic p divides t. T is called a Vandermonde set if and a super-Vandermonde set if . This (extremal) algebraic property is interesting for its own right, but the original motivation comes from finite geometries. In this paper we classify small and large super-Vandermonde sets. [Copyright &y& Elsevier]

Abstract: This paper extends the concepts from cyclic duadic codes to negacyclic codes over (q an odd prime power) of oddly even length. Generalizations of defining sets, multipliers, splittings, even-like and odd-like codes are given. Necessary and sufficient conditions are given for the existence of self-dual negacyclic codes over and the existence of splittings of 2N, where N is odd. Other negacyclic codes can be extended by two coordinates in a way to create self-dual codes with familiar parameters. [Copyright &y& Elsevier]

Abstract: In this paper we study construction algorithms for polynomial lattice rules modulo arbitrary polynomials. Polynomial lattice rules are a special class of digital nets which yield well distributed point sets in the unit cube for numerical integration. Niederreiter obtained an existence result for polynomial lattice rules modulo arbitrary polynomials for which the underlying point set has a small star discrepancy and recently Dick, Leobacher and Pillichshammer introduced construction algorithms for polynomial lattice rules modulo an irreducible polynomial for which the underlying point set has a small (weighted) star discrepancy. In this work we provide construction algorithms for polynomial lattice rules modulo arbitrary polynomials, thereby generalizing the previously obtained results. More precisely we use a component-by-component algorithm and a Korobov-type algorithm. We show how the search space of the Korobov-type algorithm can be reduced without sacrificing the convergence rate, hence this algorithm is particularly fast. Our findings are based on a detailed analysis of quantities closely related to the (weighted) star discrepancy. [Copyright &y& Elsevier]

Abstract: In this paper we construct some algebraic geometric error-correcting codes on surfaces whose Néron–Severi group has low rank. If the Néron–Severi group is generated by an effective divisor, the intersection of this surface with an irreducible surface of lower degree will be an irreducible curve, and this makes possible the construction of codes with good parameters. Such surfaces are not easy to find, but we are able to find surfaces with low rank, and those will give us good codes too. [Copyright &y& Elsevier]

Abstract: The order bound gives an in general very good lower bound for the minimum distance of one-point algebraic geometric codes coming from curves. This paper is about a generalization of the order bound to several-point algebraic geometric codes coming from curves. [Copyright &y& Elsevier]

Abstract: For a prime power q, let be the standard function in the asymptotic theory of codes, that is, is the largest asymptotic information rate that can be achieved for a given asymptotic relative minimum distance δ of q-ary codes. In recent years the Tsfasman–Vlăduţ–Zink lower bound on was improved by Elkies, Xing, and Niederreiter and Özbudak. In this paper we show further improvements on these bounds by using distinguished divisors of global function fields. [Copyright &y& Elsevier]

Abstract: For a ternary code with or 2 (mod 3), , the diversity given by is important to know about the extendability of , where stands for the number of codewords with weight i. As a continuation of [T. Maruta, Extendability of ternary linear codes, Des. Codes Cryptogr. 35 (2005) 175–190], we prove all the conjectures posed in this paper, which yield some improvements to the extendability and non-extendability of ternary linear codes. We also give the necessary and sufficient conditions for the extendability of for the case when . [Copyright &y& Elsevier]

Abstract: For any sequence over , there is an unique 2-adic expansion , where and are sequences over and can be regarded as sequences over the binary field naturally. We call and the level sequences of . Let be a primitive polynomial of degree over , and be a primitive sequence generated by . In this paper, we discuss how many bits of can determine uniquely the original primitive sequence . This issue is equivalent with one to estimate the whole nonlinear complexity, , of all level sequences of . We prove that is a tight upper bound of if is a primitive trinomial over . Moreover, the experimental result shows that varies around if is a primitive polynomial over . From this result, we can deduce that is much smaller than , where is the linear complexity of level sequences of . [Copyright &y& Elsevier]

Abstract: Difference systems of sets (DSS) are combinatorial structures that are a generalization of cyclic difference sets and arise in connection with code synchronization. The paper surveys recent constructions and open problems concerning DSS obtained as partitions of cyclic difference sets. [Copyright &y& Elsevier]

Abstract: In numerous applications involving finite fields, we often need high-order elements. Ideally we should be able to obtain a primitive element for any finite field in reasonable time. However, if the prime factorization of the group order is unknown, we do not know how to achieve the goal. We thus turn our attentions to a less ambitious problem: constructing an element of provably high order. In this paper, we survey various algorithms that find an element of high order for general or special finite fields. [Copyright &y& Elsevier]

Abstract: In this paper, we explicitly obtain the coefficient matrix arising from a linearization of Niederreiter''s factorization algorithm and analyze the complexity of setting it up. It turns out that its setup cost is linear both in the degree of a polynomial to be factored and in the size of the base field. [Copyright &y& Elsevier]

A mixed partition of PG(2n-1,q2) is a partition of the points of PG(2n-1,q2) into (n-1)-spaces and Baer subspaces of dimension 2n-1. In (Bruck and Bose, J. Algebra 1 (1964) 85) it is shown that such a mixed partition of PG(2n-1,q2) can be used to construct a (2n-1)-spread of PG(4n-1,q) and hence a translation plane of order q2n. In this paper, we provide several new examples of such mixed partitions in the case when n=2. [Copyright &y& Elsevier]

Let m2′(3,q) be the largest value of k (k for which there exists a complete k-cap in PG(3,q), q even. In this paper, the known upper bound on m2′(3,q) is improved. We also improve a number of intervals, for k, for which there does not exist a complete k-cap in PG(3,q), q even. [Copyright &y& Elsevier]

Let p be a prime, u be a linear recurring sequence of integers of order d and let S=3d2+9d/2+1. The main result of the present paper: if u is uniformly distributed mod pS, then it is uniformly distributed mod ps for all s⩾1. This solves a longstanding folklore conjecture. [Copyright &y& Elsevier]

In this paper we compute the number of curves of genus 2 defined over a finite field k of odd characteristic up to isomorphisms defined over k; the even characteristic case is treated in an ongoing work (G. Cardona, E. Nart, J. Pujola`s, Curves of genus 2 over field of even characteristic, 2003, submitted for publication). To this end, we first give a parametrization of all points in M2, the moduli variety that classifies genus 2 curves up to isomorphism, defined over an arbitrary perfect field (of zero or odd characteristic) and corresponding to curves with non-trivial reduced group of automorphisms; we also give an explicit representative defined over that field for each of these points. Then, we use cohomological methods to compute the number of k-isomorphism classes for each point in M2(k). [Copyright &y& Elsevier]

Let F be a finite field of characteristic not 2, and S⊆F a subset with three elements. Consider the collectionS={S·a+b | a,b∈F, a≠0}.Then (F,S) is a simple 2-design and the parameter λ of (F,S) is 1, 2, 3 or 6. We find in this paper the full automorphism group of (F,S). Namely, if we put U={r | {0,1,r}∈S} and K the subfield of F generated by U, then the automorphisms of (F,S) are the maps of the form x↦g(α(x))+b, x∈F, where b∈F, α : F→F is a field automorphism fixing U, and g is a linear transformation of F considered as a vector space over K. [Copyright &y& Elsevier]

In a previous paper, the authors proved that any set of representatives of the distinct 1-dimensional subspaces in the dual code of the unique linear perfect single-error-correcting code of length (qd−1)/(q−1) over GF(q) is a balanced generalized weighing matrix over the multiplicative group of GF(q). Moreover, this matrix was characterized as the unique (up to monomial equivalence) weighing matrix for the given parameters with minimum q-rank. We now relate these matrices to m-sequences (that is, linear shift register sequences of maximal period) by giving an explicit description in terms of the trace function; in this way, we show that a simple modification of our method can be used to obtain the matrices which are given by the “classical,” more involved construction going back to Berman. Moreover, further modifications of our matrices actually yield a wealth of monomially inequivalent examples, namely matrices for many different q-ranks. [Copyright &y& Elsevier]